Predicting Item Difficulties from item characteristics

Raven's (1986) progressive matrices are used to measure non-verbal intelligence: the capacity to reason by analogy, form comparisons, and think logically. The Coloured Progressive Matrices (CPM) is the 36- item version of the test used with children ages 5-12.

The complex and visual CPM items are not well understood, so we identified 14 characteristics that we thought might differentiate item difficulty (see Table 1, and Green and Kluever, 1992). Each CPM item was rated on each characteristic. Each characteristic was modelled to have its own rating scale. The 14 characteristics were then treated as agents and the 36 CPM items as objects in a Rasch analysis. The purpose of this analysis was to construct a conceptual difficulty for each CPM item based on the number and complexity of its characteristics.

CPM item characteristic map

Figure 1 shows the measures of characteristics and items. CPM items with high measures exhibited many characteristics. Characteristics with high measures were exhibited by few CPM items. Reversal (background to foreground) and B&W (colors limited to black and white) were the least observed characteristics. Only the simplest CPM items have less than 6 response Options.

Rating scale for characteristics

Figure 2 shows a map of the ratings associated with each characteristic. According to this map, a change in the number of lines and dimensions in the stem indicates less of a change in overall CPM item complexity than a change in number of elements or number of directions in the options.

To see how well these conceptual difficulty measures capture the empirical difficulties of the CPM items, the 36 CPM items were calibrated for 2 samples. The first sample was 457 1st through 5th grade children from rural southern Colorado. The second sample was 268 relatively gifted 4 through 12-year old children at an educational assessment center at the University of Denver. The results for the two samples were reassuringly similar.

After elimination of 4 CPM items as too easy or markedly misfitting, Figure 3 shows the plot of 32 conceptual item difficulties against empirically derived ones for sample 2. The correlation is .66. This implies that, after correcting for measurement error, our initial set of 14 characteristics has already explained about half the variance in item difficulties

Empirical vs. Conceptual Item Difficulties

A commonly used alternative approach to item difficulty decomposition is to regress the empirical item difficulties on the characteristics. But the item characteristics are multicollinear, so regression results are heavily influenced by the CPM-item-dependent variances in the characteristic ratings. If the characteristic rating variances were to change, even slightly (e.g., by omitting a CPM item), a different characteristic would take precedence in the regression analysis. This perplexes the researcher.

The Rasch approach, however, goes beyond multiple regression by clearly ordering the item characteristics according to their complexity. The level of successful prediction of empirical difficulties suggests that continued development of this set of item characteristics will be fruitful. The item maps provide the clarity needed to advance thought about identified and potential item characteristics and the general nature of CPM item difficulty.

Green KE & Kluever RC (1992) Components of item difficulty of Raven's Matrices. Journal of General Psychology, 119, 189-199.

Raven JC, Court JH & Raven J (1986). Coloured Progressive Matrices. London: H. K. Lewis & Co.

Table 1. Raven's Item Characteristics
Characteristic Rating value Label
In Stem of CPM Item:
Orientation 0=vertical, horizontal; 1=other Orientation
Symmetry 0=symmetrical; 1=asymmetrical Symmetry
Progression in pattern 0=no increase; 1=increase Progress-1
Dimensions in pattern 1-3=number Dimns
Lines 0=straight; 1=curved Curved
Distinct types of lines or solids 1-3=number Lines
Black and White 0=color; 1=black and white B&W
In Options of CPM Item:
Distinct options 2-6=number Options
Options contain rotation 0=no; 1=yes Rotation
Options contain reflection 0=no; 1=yes Reflection
Options contain progression 0=no; 1=yes Progress-2
Directions of options: vertical, horizontal, diagonal 1-3=number Dirns
Number of elements in the design 1-3=number Elements
Reversal between foreground and background 0 =no; 1=yes Reversal

Predicting item difficulties from item characteristics. Green KE, Kluever RC, Wright BD. … Rasch Measurement Transactions, 1994, 8:2 p.354

Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse Rasch Measurement Theory Analysis in R, Wind, Hua Applying the Rasch Model in Social Sciences Using R, Lamprianou El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Rasch Models for Measurement, David Andrich Constructing Measures, Mark Wilson Best Test Design - free, Wright & Stone
Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias Diseño de Mejores Pruebas - free, Spanish Best Test Design A Course in Rasch Measurement Theory, Andrich, Marais Rasch Models in Health, Christensen, Kreiner, Mesba Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
Rasch Books and Publications: Winsteps and Facets
Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang Statistical Analyses for Language Testers (Facets), Rita Green Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind Rasch Measurement: Applications, Khine Winsteps Tutorials - free
Facets Tutorials - free
Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

To be emailed about new material on
please enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from welcomes your comments:

Your email address (if you want us to reply):


ForumRasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website,

Coming Rasch-related Events
May 17 - June 21, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps),
June 12 - 14, 2024, Wed.-Fri. 1st Scandinavian Applied Measurement Conference, Kristianstad University, Kristianstad, Sweden
June 21 - July 19, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Winsteps),
Aug. 5 - Aug. 6, 2024, Fri.-Fri. 2024 Inaugural Conference of the Society for the Study of Measurement (Berkeley, CA), Call for Proposals
Aug. 9 - Sept. 6, 2024, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets),
Oct. 4 - Nov. 8, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps),
Jan. 17 - Feb. 21, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps),
May 16 - June 20, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps),
June 20 - July 18, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Facets),
Oct. 3 - Nov. 7, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps),


The URL of this page is